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AMH2   Integrable Systems

General Information

This page relates to the Applied Mathematics Honours course "Integrable Systems".

Lecturer(s) for this course: Nalini Joshi and Milena Radnovic.

For general information on honours in the School of Mathematics and Statistics, refer to the relevant honours handbook.


The exam paper will be available via Blackboard on Friday 10 June at 11 am. The solutions are due on Monday 13 June at 1pm and should be submitted via TurnItIn and emailed to the lecturers.

Course outline


Integrable Systems

An Honours Course in Applied Mathematics

Nalini Joshi and Milena Radnović

Weekly questions will be given below as PDF files.

Lectures delivered in the AGR will be placed here as PDF files.

The mathematical theory of integrable systems has been described as one of the most profound advances of twentieth century mathematics. They lie at the boundary of mathematics and physics and were discovered through a famous paradox that arises in a model devised to describe the thermal properties of metals (called the Fermi-Pasta-Ulam paradox).

In attempting to resolve this paradox, Kruskal and Zabusky discovered exceptional properties in the solutions of a non-linear PDE, called the Korteweg-de Vries equation (KdV). These properties showed that although the solutions are waves, they interact with each other as though they were particles, i.e., without losing their shape or speed, until then thought to be impossible for solutions of non-linear PDEs. Kruskal invented the name solitons for these solutions. Here is a picture of two solitons interacting.

two solitons (146Kb)

Solitons are known to arise in other non-linear PDEs and also in partial difference equations. These systems and their symmetry reductions are now called integrable systems. These systems occur as universal limiting models in many physical situations.

This course introduces the mathematical properties of such systems. In particular, we will study their solutions, symmetry reductions called the Painlevé equations and their discrete versions. It focuses on mathematical methods created to describe the solutions of such equations and their interrelationships. More details about the course, including course objectives and outcomes and details about the assessment and exam can be read on the Information Sheet (PDF).

  • References:
    • M. J. Ablowitz and H. Segur, Solitons and the inverse scattering transform, SIAM, Philadelphia, USA, 1981.
    • M. J. Ablowitz and P.A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press, Cambridge, UK, 1991.
    • P.G. Drazin and R.S. Johnson, Solitons: an introduction, Cambridge University Press, Cambridge, UK, 1989.
    • M. Noumi, Painlevé equations through symmetry, American Mathematical Society, Providence, R.I., USA, 2004.

  • Interesting Links on Solitons:
    • An account of John Scott Russell's discovery of "that singular and beautiful phenomenon, which I have called the wave of translation."
    • A modern attempt by mathematicians to recreate Scott Russell's wave in the Union Canal near Edinburgh.
    • The Wikipedia page on Solitons (whose first and second definitions are still not correct).
    • The Wikipedia page on the Korteweg-de Vries equation.

    Last modified: 20 February 2016 by

Online resources

Monday lecture notes Wednesday lecture notes Assignments
Week 1 Lecture 1 Lecture 2 Assignment 1 / Solution
Week 2 Lecture 3 Lecture 4 Assignment 2 / Solution
Week 3 Lecture 5 Lecture 6 Assignment 3 / Solution
Week 4 Lecture 7 Lecture 8 Assignment 4 / Solution
  Semester break
Week 5 Lecture 9 Lecture 10 Assignment 5 / Solution
Week 6 Lecture 11 Lecture 12 Assignment 6 / Solution
Week 7 Lecture 13 Lecture 14 Assignment 7 / Solution
Week 8 Anzac Day Lecture 15 Assignment 8 / Solution
Week 9 Lecture 16 Lecture 17 Assignment 9 / Solution
Week 10 Lecture 18 Lecture 19  
Week 11 Lecture 20 Lecture 21 Assignment 10 / Solution
Week 12 Lecture 22 Lecture 23  


Last revised 31/05/16

10am Lecture
(Wks 1-7,9-12)
(Wks 1-12)